Rotemberg in a DCP/LCP layout

[HelloDynare]

Hello everyone,

Recently, I’m rethinking a very inspiring dialogue in:

Rotemberg term in the market clearing condition

Hello everyone, I have a question regarding Rotemberg adjustment term in the market clearing condition. Let’s set our baseline model be Gali and Monacelli (2005) or the relevant chapters in Gali (2015). According to their formulation, we can write the market clearing condition as follows: \displaystyle Y_t(i) = C_{H,t}(i)+X_{t}(i) \displaystyle Y_t(i)=(1-\nu)\left(\frac{P_{H,t}(i)}{P_{H,t}}\right)^{-\epsilon}\left(\frac{P_{H,t}}{P_t}\right)^{-\eta}C_t+\nu\left(\frac{P_{H,t}(i)}{P_{H,t}}\rig…

especially, the reply below that I posted one year ago

Rotemberg term in the market clearing condition

Mm…you may want to ask among all “micro-level conditions” that you listed, which will no longer hold, if taking the Rotemberg frictions into account? (1-\gamma)\frac{\delta}{2}\left( \pi_{Ht}-\pi_H\right)^2Y_{Ht} and \gamma\frac{\delta}{2}\left( \pi^*_{Ht}-\pi^*_H\right)^2Y^*_{Ht} that have Y_{Ht} and Y_{Ht}^* terms. My thought: Nothing wrong, except: \displaystyle \mathcal{F}_t = C_t + (1-\gamma)\frac{\delta}{2}\left( \pi_{Ht}-\pi_H\right)^2Y_{Ht}+\gamma\frac{\delta}{2…

Since I know @DoubleBass read some DCP literature, @valerio88 nicely answered @littlemacro and my naive questions last year, posts of @dgroll were exactly what I thought, and I’m always deeply indebted to @jpfeifer , hope you and other dynare users may be interested in these questions Welcome all comments

Allow me to write my questions in next posts. Please be patient, thx!

[HelloDynare]

The figure is captured from the paper:
The Macroeconomics of Border Taxes, Gita Gopinath et al. NBER Macro Annual 2018

The Macroeconomics of Border Taxes.pdf (931.8 KB)

I’m particularly eager to know what I should write in the red rectangle, if I reformulate the problem in a Rotemberg setting, instead of Calvo as shown in the paper.

[HelloDynare]

(In the following, notations slightly vary from the paper, but the meaning keeps the same.)

For the Home Producer, authors said:

Y_t(\omega)=A_tL^{1-\alpha}_t(\omega)X^{\alpha}_{t}(\omega)
\mathcal{MC}_t=\kappa A^{-1}_tW^{1-\alpha}_tP_t^{\alpha} , where \kappa\equiv \frac{1}{\alpha^{\alpha}(1-\alpha)^{1-\alpha}}
P_tX_t=\alpha \mathcal{MC}_t Y_t, and W_tL_t=(1-\alpha)\mathcal{MC}_tY_t
So \mathcal{MC}_tY_t = P_tX_t+W_tL_t.

[HelloDynare]

Now I modify the pricing block under the Rotemberg.
The intermediate-good producer wants to max the profit in the home market:
\displaystyle\max_{\{P_{HHt}(\omega)\}} \mathbb{E}_t \sum_{t=0}^{\infty} \Lambda_{0,t} \left\{ \left[ P_{HHt}(\omega) - \mathcal{MC}_{t} \right] Q_{HHt}(\omega) -\frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2 P_{HHt}Q_{HHt} \right\}
, and in the foreign market:
\displaystyle\max_{\{P^*_{HFt}(\omega)\}} \mathbb{E}_t \sum_{t=0}^{\infty} \Lambda_{0,t} \left\{ \left[ \mathcal{E}_tP^*_{HFt}(\omega) - \mathcal{MC}_{t} \right] Q^*_{HFt}(\omega) -\frac{\xi_P}{2}\left[\frac{P^*_{HFt}(\omega)}{ P^*_{HF,t-1}(\omega)}-\bar{\Pi}^*_{HF}\right]^2 \mathcal{E}_tP^*_{HFt}Q^*_{HFt} \right\}.
After we impose symmetric equilibrium, all (\omega) will be removed. We can obtain profits per period:
prof_{HHt}=\left(P_{HHt}- \mathcal{MC}_t\right) Q_{HHt}-\frac{\xi_P}{2}\left(\Pi_{HHt}-\bar{\Pi}_{HH}\right)^2 P_{HHt}Q_{HHt},
prof^*_{HFt}=\left(\mathcal{E}_tP^*_{HFt}- \mathcal{MC}_t\right) Q^*_{HFt}-\frac{\xi_P}{2}\left(\Pi^*_{HFt}-\bar{\Pi}^*_{HF}\right)^2 P^*_{HFt}Q^*_{HFt},

[HelloDynare]

On the other hand, for the Home Bundler, suppose the final goods are \mathcal{F}_t. Authors said:
\mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{Q_{HHt}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{Q_{FHt}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}.
Note, according to the arrows, it’s Q_{FHt} rather than Q^*_{HFt}. Also,
\mathcal{F}_t=C_t+X_t+G_t

[HelloDynare]

Here come to my questions!

In Rotemberg, we can directly write out:

Y_t = Q_{HHt} + Q^*_{HFt} + \frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2 Q_{HHt} + \frac{\xi_P}{2}\left[\frac{P^*_{HFt}(\omega)}{ P^*_{HF,t-1}(\omega)}-\bar{\Pi}^*_{HF}\right]^2 Q^*_{HFt}

BUT after multiplying \mathcal{MC}_t on both sides,
\mathcal{MC}_tY_t = P_tX_t+W_tL_t=\mathcal{MC}_t\left(Q_{HHt} + Q^*_{HFt}\right)
+ \mathcal{MC}_t \left\{ \frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2 Q_{HHt} + \frac{\xi_P}{2}\left[\frac{P^*_{HFt}(\omega)}{ P^*_{HF,t-1}(\omega)}-\bar{\Pi}^*_{HF}\right]^2 Q^*_{HFt} \right\}

The first line of the equalities will cancel the relevant terms in prof_{HHt} and prof^*_{HFt}. HOWEVER the second line is \mathcal{MC}_t*Rotemberg, not the pure Rotemberg, hence has no chance to cancel terms in prof_{HHt} and prof^*_{HFt}.

It implies the aggregate country budget constraint will be wrong!

Furthermore, do I need change \mathcal{F}_t to:
\mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{\tilde{Q}_{HHt}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{\tilde{Q}_{FHt}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}
in which, because of the Rotemberg friction / deadweight loss,
\tilde{Q}_{HHt}= \displaystyle\frac{Q_{HHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2}
and
\tilde{Q}_{FHt}= \displaystyle\frac{Q_{FHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{FHt}(\omega)}{ P_{FH,t-1}(\omega)}-\bar{\Pi}_{FH}\right]^2}
??

[jpfeifer]

1. Are you doing higher order solutions or trend inflation? If not, most of the differences in the budget constraint do not matter, because the quadratic costs will drop out.
2. Because Rotemberg pricing involves actual resource costs, you need to decide in which units these costs are paid. Here, you have some leeway. It’s up to you to have them paid in different or identical units in different sectors. You just have to make sure to include appropriate relative prices when adding them up.

[HelloDynare]

In fact, I believe I have already found out the solution
Thank you for replying me , and you prevent my embarrassment from answering my own questions (s.t. all posts in one topic were solely from me )
Please be tolerant I asked the same questions to another professor, waiting for his reply. In any circumstances, I’ll come back and make out my thought in few days.

[alexfra1709]

It is a great post, I also interest in this topic. Did you get the answers on this ? Thank you.

[HelloDynare]

Hi @alexfra1709 ,

Thank you for your attention! I’m happy someone also shows the interests in this topic

I found the solution in the configuration of an old paper:

https://www.sciencedirect.com/science/article/pii/S0022199602000570

The key point is actually there are two market clearing conditions, the one for intermediate goods and the other for final goods.

@jpfeifer

Because Rotemberg pricing involves actual resource costs, you need to decide in which units these costs are paid. Here, you have some leeway.

As you have already seen, if Rotemberg terms are added into Y_t, the country budget constraint will be wrong. So try to annex them into \mathcal{F}_t:

\mathcal{F}_t = C_t + X_t +G_t +\frac{\xi_P}{2}\left(\Pi_{HHt}-\bar{\Pi}_{HH}\right)^2p_{HHt}Q_{HHt}+\frac{\xi_P}{2}\left(\Pi^*_{HFt}-\bar{\Pi}^*_{HF}\right)^2p^*_{HFt}Q^*_{HFt}.

@jpfeifer

You just have to make sure to include appropriate relative prices when adding them up.

Yes, p_{HHt} and p^*_{HFt} guarantee this.

In this sense, we adjust the condition for Y_t as follows:
Y_t = Q_{HHt}+Q^*_{HFt}.

You could now sum up all relevant budget constraints together, the ultimate country-wise budget constraint is indeed what Prof. Pfeifer commented:

@jpfeifer

most of the differences in the budget constraint do not matter, because the quadratic costs will drop out

[HelloDynare]

@HelloDynare

Furthermore, do I need change \mathcal{F}_t to:
\mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{\tilde{Q}_{HHt}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{\tilde{Q}_{FHt}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}
in which, because of the Rotemberg friction / deadweight loss,
\tilde{Q}_{HHt}= \displaystyle\frac{Q_{HHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2}
and
\tilde{Q}_{FHt}= \displaystyle\frac{Q_{FHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{FHt}(\omega)}{ P_{FH,t-1}(\omega)}-\bar{\Pi}_{FH}\right]^2}
??

I find one recent paper goes along this line as well. But the treatment is more complicated than my previous post.

[alexfra1709]

This is great. Thanks for your effort. Looks like it is easier to implement than Calvo setup.